Polynomial Functions

A polynomial function uses a polynomial expression as an input-output rule. Its coefficients determine a continuous graph whose zeros, end behaviour and turning points can be studied through its algebraic structure.

The Polynomials page introduces terms and operations. Here the focus is how factors and the leading term describe a function’s graph.

On this page (8 sections)

Polynomial-function form

f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0

For a non-zero polynomial, $n$ is a non-negative integer, the coefficients are real constants, and $a_n\ne0$ . Its degree is $n$ . A polynomial formula is defined for every real input, although an application may restrict the domain.

A non-zero constant polynomial has degree zero. The zero polynomial is the special rule $f(x)=0$ ; its degree is not assigned an ordinary non-negative value here. Expressions such as $1/x$ and $\sqrt x$ are not polynomial expressions in $x$ .

Zeros and factors

A zero is an input $r$ for which $f(r)=0$ . For $f(x)=x^3-4x$ , factor the common term and then the difference of squares:

x^3-4x=x(x^2-4)=x(x-2)(x+2)

x(x-2)(x+2)=0\quad\Longleftrightarrow\quad x=-2,\ 0,\ 2

Three simple zeros of the same cubic

$y=x^3-4x$

The graph crosses at

x=-2

0

and

2

Diagram explained

All three factors have multiplicity one, so the sign changes at each zero. The positive leading cubic term determines the fall to the left and rise to the right beyond this finite window.

Exact reference values for this graph
Input	Output
$-3$	$-15$
$-2$	$0$
$-1$	$3$
$0$	$0$
$1$	$-3$
$2$	$0$
$3$	$15$

The intercept points are $(-2,0)$ , $(0,0)$ and $(2,0)$ . Factoring identifies these exact coordinates; a plotting window alone is not a reliable way to count all zeros of an arbitrary polynomial.

End behaviour and the leading term

The highest-degree term dominates the end behaviour. For non-constant real polynomials, use both the degree’s parity and the leading coefficient’s sign.

End behaviour of a non-constant polynomial
Degree	Leading coefficient	Left end	Right end
Even	Positive	Rises	Rises
Even	Negative	Falls	Falls
Odd	Positive	Falls	Rises
Odd	Negative	Rises	Falls

For $x^3-4x$ , the leading term is $x^3$ , so values fall without bound to the left and rise without bound to the right. The lower-degree term still matters near finite inputs; the leading term does not determine every part of the curve.

Multiplicity changes how a graph meets the axis

If $f(x)=(x-r)^m q(x)$ and $q(r)\ne0$ , then $r$ has multiplicity $m$ . Near $r$ , the sign of $q(x)$ stays fixed. The factor $(x-r)^m$ changes sign for odd $m$ , but not for even $m$ .

The graph therefore crosses the axis at an odd-multiplicity zero and touches without crossing at an even-multiplicity zero.

(x-1)^2(x+2)=x^3-3x+2

Here $x=1$ is a double zero and $x=-2$ is a simple zero. Near 1, the square stays non-negative and $x+2$ stays positive, so the curve touches from above. At -2, the linear factor changes sign.

Degree constrains but does not determine the entire graph

A non-zero polynomial of degree $n$ has at most $n$ distinct real zeros. A polynomial of degree $n\ge1$ has at most $n-1$ turning points. These are upper bounds, not predictions that every available zero or turn must occur.

For example, $x^2+1$ has degree 2 but no real zeros. The cubic $x^3$ has just one distinct zero and no turning point, even though the cubic degree allows more. The zero polynomial is excluded from the finite-zero bound because every real input is a zero.

Practice questions

Factor $x^{3}-4x$ .
What are the real zeros of $x^{3}-4x$ ?
What mainly determines a polynomial’s end behaviour?

Check your answers

$x^3-4x=x(x^2-4)=x(x-2)(x+2)$ .
The factors vanish at $x=0$ , $x=2$ and $x=-2$ . The distinct real zeros are $-2,0,2$ .
The degree’s parity and the sign of the leading coefficient determine the two end directions for a non-constant polynomial.

Sources and further reading

OpenStax: Zeros, multiplicity and polynomial graphs. Definitions, conventions and related examples.
OpenStax Precalculus 2e. Functions, transformations, inverse functions and function families.
Wolfram Language: FunctionDomain. Domain conventions and symbolic domain checks.
Wolfram Language: FunctionRange. Range conventions and symbolic range checks.